International Journal of Computational and Mathematical Sciences 2;1 © www.waset.org Winter 2008
Evolutionary Dynamics on Small-World Networks
Jan Rychtář and Brian Stadler
probability. However, fixation probability does not have to
depend on r this way, [4].
The fixation probability for special graphs such as a complete graph, a square lattice, and a hexagonal lattice, is given
by
1 − 1/r
,
(1)
=
1 − 1/rN
Abstract—We study how the outcome of evolutionary dynamics on
graphs depends on a randomness on the graph structure. We gradually
change the underlying graph from completely regular (e.g. a square
lattice) to completely random. We find that the fixation probability
increases as the randomness increases; nevertheless, the increase is
not significant and thus the fixation probability could be estimated
by the known formulas for underlying regular graphs.
Keywords—evolutionary dynamics, small-world networks.
where N is the size of the population, [4].
The dynamics on small-world networks, [9], and random
graphs, [10], [11], were studied. A small-world network is a
random graph where most vertices are not neighbors to each
other, yet the majority of vertices can be reached from any
other vertex in a relatively short path through the graph. Social
networks, the Internet and national power grids all exhibit the
characteristics of small-world networks, [9].
We produced a small-world network starting from a regular
graph structure by randomly rewiring a gradually increasing
percentage of edges. We then looked at how the fixation
probability of mutants depends on the following three
parameters:
• fitness r of a mutant,
• percentage of the changed vertices c,
• and the underlying original regular graph structure.
I. I NTRODUCTION
Evolutionary dynamics has been traditionally studied in
infinite homogenous, [1], infinite spatial, [2], populations.
Recently, the dynamics was studied in finite, [3], and spatially
structured populations (i.e. graphs), [4], [5, Chapter 8], [6],
[7].
A graph can capture spatial, social and/or other structures
of the population. Graphs can represent nearly everything
we encounter in life, cities interconnected by highways, the
national power grid, ecological structures, the social networks
within which we communicate, etc. Many of these naturally
occurring graphs can be modeled as ’small-world’ networks,
[8].
For the purpose of the dynamics, every vertex of a graph
represents an individual. Individuals can place offspring into
adjacent vertices. How often an individual is selected for
reproduction is proportional to individual’s fitness. The higher
the fitness, the more likely it is that the individual will be
selected.
For a given graph, it is important to know what happens
when a mutant with a fitness r is introduced into a population
of otherwise equal individuals of fitness 1. There are three
possible scenarios:
1) A mutant population will eventually spread through
the entire graph and will replace all of the original
inhabitants,
2) The original population will recover from the mutant
invasion and will eventually wipe out all mutants,
3) Both mutants and the original inhabitants stay in the
population for an infinite amount time.
We are interested in the fixation probability of a mutant,
i.e. a probability of the scenario 1). Note that since the
dynamics is not deterministic, the probability is never 1, no
matter how advantageous is the mutant. Also, it seems intuitive
that the higher the fitness r of a mutant, the higher its fixation
II. M ATHEMATICAL BACKGROUND
Only undirected graphs, G = (V, E), where V =
{0, 1, 2, · · · , N − 1} is the set of vertices and E is the set
of edges, were considered.
We regarded the dynamics as a Markov chain, [12]. If the
mutants already inhabit vertices in the set C ⊂ V , then in the
next step the mutants will inhabit either
1) a set C ∪ {j}, j ∈ C, provided a) a vertex i ∈ C was
chosen for a reproduction and b) it placed its offspring
into j; or
2) a set C \ {i}, i ∈ C, provided a) a vertex j ∈ C was
selected for a reproduction and b) it placed its offspring
into a vertex i; or
3) a set C, provided an individual from C (V \ C) replaces
another individual from C (V \ C).
The states ∅ and V are the absorbing points of the dynamics.
To determine the transitions probabilities of the above Markov
chain, we have to determine a) the probability that a given
vertex will be selected for a reproduction and b) the probability
that, once selected, it places its offspring into another vertex.
Let an individual i have a fitness fi ∈ {1, r}, where fi = r
means that i is a mutant. To be selected with a probability
proportional to its fitness means that an individual i is selected
for a reproduction with probability
fi
(2)
si = N −1 .
j=0 fj
J. Rychtář is with the Department of Mathematics and Statistics University
of North Carolina at Greensboro Greensboro, NC 27402, USA, email:
[email protected].
B. Stadler is with the Department of Computer Science University of
North Carolina at Greensboro Greensboro, NC 27402, USA, email: [email protected].
The research was supported by the NSF grant 0634182. B. Stadler was
supported by the UNCG Undergraduate Research Assistantship.
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International Journal of Computational and Mathematical Sciences 2;1 © www.waset.org Winter 2008
We represented the graph by a matrix W = (wij ), where
wij = 0, i and j are not connected, and wij = ei −1 , if i and
j are connected, where ei is the number of edges coming out
of a vertex i. In this notation, wij is the probability that (if
selected for a reproduction) an individual i places an offspring
into a vertex j.
Let PC denote the probability of mutant fixation (i.e. mutants reaching the state V ) provided mutants currently inhabit
a set C. The rules of the dynamics yield
r
wij PC∪{j}
PC
=
i∈C j∈C
r
wij +
i∈C j∈C
•
wji
j
∈C i∈C
wji PC\{i}
j∈C i∈C
+ r
wij +
wji
i∈C j∈C
be able to quickly manipulate these graphs,
generate random numbers.
C++ was chosen as the language to handle the simulation. It was primarily chosen due to available C++ libraries
from http://www.boost.org, which can handle the requirements
listed above. Below are the specific libraries used in development of the program:
• Boost date time libraries - to aid in seeding the random
number generator,
• Boost graph libraries - for creation of undirected graphs
and tracking the properties of each vertex,
• Boost random libraries - for creation of large amounts
of near non-deterministic numbers. Specifically, the
mt19937 generator was used which has a cycle of
219937 − 1.
•
(3)
B. Evolutionary dynamics
j∈C i∈C
Below is a pseudo code representing the evolutionary dynamics on a given graph:
with the “boundary conditions”
P∅
PG
=
=
0,
1.
while (not (graph fully mutated or without mutants)) do
select (a base vertex for reproduction)
find (all neighboring vertices of base vertex)
select (a neighboring vertex to be replaced)
replace (neighboring vertex by offspring of base vertex)
end
The above system (3) of linear equations is unfortunately
very large and very sparse. From any state C, one can go to
at most N other states, i.e. each row of the transition matrix
contains at most N non-zero elements. Yet, even the simplest
graphs like a circle or a line contain of the order of N 2
states that can be reached starting from a single mutant in
any position i ∈ V . Moreover, once the graph is not a circle
and not a tree, (3)consists of of the order of 2N equations,
[13].
As a result, (3) can be seldom solved. In [4], authors abandoned (3) completely by restricting themselves to isothermal
graphs, i.e. graphs satisfying
N
−1
wji =
j=0
N
−1
The core of the process is the selection of the base vertex
for reproduction. An individual labeled i is chosen for a
reproduction with probability si given by (2). Once a base
vertex is selected, all adjacent vertices are found. If a given
base vertex has M neighbors, a specific neighbor is selected
with a probability 1/M .
C. Creation of small-world networks and random graphs
The procedure for creation of small-world networks follows
[9]. We randomly rewire a given percentage of edges in
the original graph. Loop back and parallel edges were not
allowed. We also disregarded all disconnected graphs that
could possibly result in this procedure (a disconnected graph
has fixation probability 0 because mutants can never spread
outside the component they were originally introduced to).
Below is pseudo code representing the creation of these
graphs:
wji , i = i .
j=0
For such graphs, (3) reduces due symmetries into a one
dimensional random walk that corresponds to the Moran
process, [14]. Examples of isothermal graphs are complete
graphs, square lattice, hexagonal lattice, etc.
In [15], authors solved (3) for special values of r ≈ 1 and
r >> 1.
Our goal is to attack the dynamics for general r and general
graphs. We used a Markov Chain Monte Carlo method, [16],
in order to get the estimate of P{i} , i ∈ V and then calculate
the fixation probability
=
while (less than c percent of original edges changed) do
select(random base vertex)
find (all neighboring vertices of base vertex)
select (one neighboring vertex)
destroy(edge between base and neighbor vertex)
select(new random vertex)
create(edge between base and new random vertex)
end
check(connected graph)
N −1
1 P{i} .
N i=0
III. MCMC
A. Computing background
All selections here are not fitness related. If we had N
vertices, one particular vertex was chosen with probability
1/N . If we had M neighboring vertices to choose from, one
The MCMC simulation of the evolutionary dynamics had
to:
• represent complicated graph structures,
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International Journal of Computational and Mathematical Sciences 2;1 © www.waset.org Winter 2008
a)
b)
Fig. 2. Fixation probabilities for square lattice with 49 vertices and a relative
mutant fitness 1.1.
c)
1, . . . , 20} was produced using the above rewiring algorithm to
randomly rearrange the edges of the base graph. Graph Gi had
i·5% of rewired edges. We then ran the MCMC simulation on
each such graph and recorded the results. In the beginning, we
performed 106 runs for each graph and every mutant fitness
r. Since 106 runs took anywhere from 10 to 24 hours for one
graph, the worst case time scenario was about 200 days using
a single computer. No large scale computing environment was
available.
After completing results for r = 1.1, r = 1.5 and r = 2
we looked at all of the results and it was determined that
the results would not differ significantly if we stopped the
simulations after 105 runs. Thus, the remaining results were
generated using this smaller amount of runs.
d)
IV. R ESULTS
There are two fundamental results of the simulations:
e)
1) the mutants do perform better in the changed graphs
than in the original regular graphs;
2) the fixation probability of mutants in rewired graphs
is never significantly above the value predicted by the
formula (1).
f)
Fig. 1. Increase of randomness. a) Regular square lattice (c = 0), b) c =
0.05, c) c = 0.25, d) c = 0.5, e) c = 0.75, f) c = 1.
Generally, the more edges were rewired the better the
mutants performed. See Figure 2, where results are plotted
for mutant fitness r = 1.1 and compared to (1). Results for
other mutant fitness in the range 1.1 ≤ r ≤ 2 were analogous.
As one can see from Figure 2, it is not always true that
the more edges rewired the better the mutants perform. There
are some minor fluctuations, cases when a slight increase
of randomness actually decreased the fixation probability of
mutants. The fluctuations flatten out as we consider different
rewirings and taking averages over the rewirings.
We observed that the fixation probability of mutants in
rewired graphs is never significantly above the value predicted
by the formula (1). As we can see from Figure 3, one can
effectively estimate the fixation probability in any random
rewiring. The fixation probability was never more than .041
above the value predicted by the formula (1) (this was the
case of r = 1.5); respectively never more than 17% of the
particular vertex was selected with the probability 1/M . The
selection of the new random vertex had to be a bit more
sophisticated. To assure that the resulting graph does not have
loops or multiple edges, we had to exclude the base vertex
itself as well as all vertices that were neighboring with the
base vertex at the time of its selection. Figure 1 shows how a
regular square lattice changes its structure as more and more
edges are rewired.
D. Simulations
Evolutionary dynamics with mutant fitness r ∈ {1 +
j/10; j = 1, 2, . . . , 10} were simulated. We did not simulate
r > 2 extensively since it is not biologically too relevant to
have a mutant with a fitness significantly higher than the rest
of the population.
For each fitness value a set of 20 graphs, {Gi , i =
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International Journal of Computational and Mathematical Sciences 2;1 © www.waset.org Winter 2008
For smaller graphs, such as those representing computer
networks, or power grids, it may be possible to predict how
a computer virus will spread through a network or whether
or not the failure of one power station will cause a cascading
effect and in turn shut down the entire grid.
Further work needs to be done. If one wants to suppress a
virus in a network what graph will best do this? The theoretical
results from [4] as well as numerical simulations we performed
suggest that the greater the variation of various parameters of
the graph the better the mutants do. Among the characteristics
to consider are vertex temperature (see [4] for the definition),
path length, and clustering.
ACKNOWLEDGMENT
Fig. 3. Difference in fixation probabilities for square lattice with 49 vertices,
r ∈ [1.1, 2] and c ∈ [0, 1].
The authors would like to thank the editors of the journal
for valuable suggestions that helped to increase the quality of
the paper.
R EFERENCES
Fig. 4.
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Slopes of the trend lines as a function of mutant’s fitness.
predicted value (the case of r = 1.1), typically 9% above the
predicted value.
We also noticed a behavior that we cannot explain. As
already noted, the more edges we changed, the higher the
fixation probability of mutants. As seen in Figure 2, the
dependence is linear, i.e. for a given base graph, the fixation
probability for a graph with c edges changed is given by
(c) = (0) + a + bc.
Figure 4 shows how the slope of the trend lines depends on
the fitness r. The slopes are increasing for r between 1.1 and
1.6 (from 0.01 to 0.025) and then it sharply goes down to
roughly 0.015 and stays around this value. Although it seems
plausible that the effect of the graph structure will be weaker
as mutants’ fitness gets larger, we still do not know what is
the exact cause of this sharp decline.
V. D ISCUSSION
We have studied the evolutionary dynamics on small world
networks. We have shown that the mutants do perform better
in the small-world network graphs than in regular graphs. The
fixation probability is never below the level for a mutant with
the same fitness in the base regular graph. However, it is never
significantly above the level.
4